Regarding ComplexExpand, your expressions for d, integral powers of
Abs, are already in expanded forms acceptable to ComplexExpand. You
want alternate forms without Abs. Tell ComplexExpand that,
ComplexExpand[d, TargetFunctions -> {Re, Im}]
and it will give you the answers you want.
Each new application of Simplify or FullSimplify is an adventure :-)
> Could somebody explain why Mathematica does not return the same
> expression in the following cases. ...